It's a heck of a lot easier to remove it all in one piece than it is to try to laboriously patch all the holes that removing it piecemeal would entail. (It also lets you more easily run the wires to your new undercabinet lights.) You can just replace it with a large single patch of either drywall or cement board, prime it, and be done and ready for the new backsplash on top of it.

The description of how to do it sounds harder than it actually is. You will need to remove the drywall entirely and replace it with cement board It's easier than replacing it with more drywall as you don't have to mud it flat and the fiberglass mesh tape just needs the lightest skim coat of thinset to set it. To be able to do that job, you will need a heavy bladed utility knife, a flat prybar, a hammer, a T-square, a cordless drill of at least 12v, stainless steel screws, fiberglass mesh tape and cement board. The cement board is 36" x 60" and you will need a way to haul it home. Screw the cement board to the studs, getting the screw heads slightly indented into the surface of the board. Spray the cejent board with water and mix up a small batch of the thinset and use a flat bladed drywall tool to smooth it lightly over the seam. Imbed the fiberglass mesh tape into that smoothly and let dry. To be able to install the tile, you will need the dry mix thinset, a bucket to mix it in, a large paint stir stick to mix it, a notched trowel to apply it, the correct amount of tile, tile spacers and you will need a tile saw. You can purchase an inexpensive one for around $80 and have it for as long as you need it to be able to little bit the project to death. If you rent, you can expect to pay that $80 for a 24 hour rental, so really, it makes more sense to buy the saw and then you can take as long as you need to and you'll have it for other projects. Once the thinset has cured, you will need to grout it. You will need the powdered grout of your choice, another bucket to mix it in, another stir stick, a grout float, a couple of sponges and a bucket of water. Some microfiber cloths to polish off the rest of the residue after it's dried is also a good idea. You will want to seal the grout, so you will need an applicator bottle and grout sealer. Getting the items you need is half the battle. It's also a good idea to practice on a garage sale find and create yourself a small patio table first while you get the hang of how to mix up the thinset and grout.

three leaf clover: Check your countertops for level. Install plastic non-compressive shims if not. Place a continuous shim the thickness of the grout line on the counter. I've used 2 rips of 3/4" wide plastic laminate with great success. You may want to dab some hot melt to hold it in place. Adhere your tile to the wall, starting by placing the first row on the shim. Tile as high as necessary to get where you need to be. Grout the tile, then pull the shim and push color-match silicone into the gap.

FWIW, after too many attempts to save drywall when removing tile I now just start by cutting the drywall and tile out completely. It's way faster and easier to patch in a new piece of drywall than to fuss with careful removal of the tile and subsequent repair of the damage. This has the side benefit if you're re-tiling of being able to put in a more appropriate backer board in place of the drywall. It is disappointing when renovation doesn't go to plan, but more common than not in my experience.